“One funny thing was when a bunch of players — there were five at the time — started interrogating me about just how it is this NPC maintains power and authority over this tiny town. And I’m like…tradition? And they start pushing back on that. So I start digging, digging, digging into the material at hand. It’s not addressed (I don’t think) and then I’m like…I could make up an answer but that’ll most likely contradict something else later on. It happens with depressing regularity when I try to use pre-made stuff.”

I see three possible causes: Complicated scenario, poorly presented information, or lack of skills/wrong approach on your part.

1. The situation presented by the adventure might be complicated. I usually have pretty easy time internalizing modules. For me, a difficult one is, for example, Melford murder (free on Dragonsfoot, I think), because there are a large number of NPCs with particular agendas, relationships and knowledge of each other. Furthermore, players can meet any of them at any point, so it is difficult to know what to prepare for. Maybe you know better how to process this with relationship maps and what not, as this kind of content is more abundant in many Forge games.
Another complicated thing might be intricate special effects in rooms or dungeons with strange geometry (hypercubes being famous) or with lever to change orientation or water level or gravity and what not. These might be difficult to understand because the phenomenon is inherently complicated. Good presentation can simplify it, to a point.
Third thing might be that the thing is big. Since movement options within dungeons are often limited, I try to read carefully the areas where it is easy to end up in, and the obstacles to moving to other places, so that if the players start planning something strange I can read up on the next room while they are figuring out what to do about the pit with a wall of force above it to prevent jumping.

2. If the information is poorly presented, one has to do more work to get it out, or give up.

3. I have no idea how you approach preparing a dungeon game.
I have usually read the dungeon before and placed it in my game world. I will also have processed it: figured out which, if any, ancient culture it is related (in my head), drawn out rumours from the adventure background and NPC descriptions (inserted them into rumour tables of nearby settlements and then forgotten them), expanded the random encounter table to include most creatures living in the place (and forgotten all about it), added references to nearby other random encounter tables and the regional encounter table (and forgotten all about it), and considered how to express mechanically the parts of the adventure which were written for a different rules system (in my head or forgotten). The processing typically requires a reread or two of the relevant parts – background knowledge for rumours and which creatures are mobile and how many there are for random encounters.
Then, at some point, the players might get there. If I know of this ahead of time, I will reread the adventure so that I check the map when reading the room descriptions to get a better idea of what is where and which rooms they will be entering first. During play I keep doing this when the players are planning or speculating if I feel uncertain about the contents of the next room; their plans and speculations are not really my concern, anyway, their actions are.

This works for me. If you think this is a problem for you and you want to get better, maybe find a local OSR referee to mentor you. Or talk with someone online.